Answer to old chestnut: convoluted states I
This happens at the border between Kentucky, Missouri, and Tennessee.
The boundary between Kentucky and Tennessee is a line of latitude, but the boundary between both these states and Missouri is the Mississippi River. It so happens that the river meanders quite a bit in this region, such that it crosses that particular line of latitude three times. As a result, a little peninsula of Missouri cuts all the way across Kentucky and down into Tennessee, and a small roundish portion of radius about 5 miles at the end of Kentucky is entirely cut off from the rest of the state.
In order to have three states meet in three places, it is necessary for one of them to be disconnected, as Kentucky is here. It is possible for three fully-connected states to meet in two places, if one stretches around so that it meets the border between the other two states at both ends, but three meeting points is impossible for three fully-connected states.